Download e-book for iPad: Integration of functions of single variable by Hardy. G. H. (Godfrey Harold). 1877-1947.

By Hardy. G. H. (Godfrey Harold). 1877-1947.

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P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

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Integration of functions of single variable by Hardy. G. H. (Godfrey Harold). 1877-1947.


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